20,301 research outputs found
Deterministic modal Bayesian Logic: derive the Bayesian inference within the modal logic T
In this paper a conditional logic is defined and studied. This conditional
logic, DmBL, is constructed as a deterministic counterpart to the Bayesian
conditional. The logic is unrestricted, so that any logical operations are
allowed. A notion of logical independence is also defined within the logic
itself. This logic is shown to be non-trivial and is not reduced to classical
propositions. A model is constructed for the logic. Completeness results are
proved. It is shown that any unconditioned probability can be extended to the
whole logic DmBL. The Bayesian conditional is then recovered from the
probabilistic DmBL. At last, it is shown why DmBL is compliant with Lewis'
triviality.Comment: Second revision of: Definition of a Deterministic Bayesian Logi
Deterministic Bayesian Logic
In this paper a conditional logic is defined and studied. This conditional
logic, Deterministic Bayesian Logic, is constructed as a deterministic
counterpart to the (probabilistic) Bayesian conditional. The logic is
unrestricted, so that any logical operations are allowed. This logic is shown
to be non-trivial and is not reduced to classical propositions. The Bayesian
conditional of DBL implies a definition of logical independence. Interesting
results are derived about the interactions between the logical independence and
the proofs. A model is constructed for the logic. Completeness results are
proved. It is shown that any unconditioned probability can be extended to the
whole logic DBL. The Bayesian conditional is then recovered from the
probabilistic DBL. At last, it is shown why DBL is compliant with Lewis
triviality.Comment: Fourth version. A sequent formalism is use
Deterministic modal Bayesian Logic: derive the Bayesian within the modal logic T
In this paper a conditional logic is defined and studied. This conditional
logic, DmBL, is constructed as close as possible to the Bayesian and is
unrestricted, that is one is able to use any operator without restriction. A
notion of logical independence is also defined within the logic itself. This
logic is shown to be non trivial and is not reduced to classical propositions.
A model is constructed for the logic. Completeness results are proved. It is
shown that any unconditioned probability can be extended to the whole logic
DmBL. The Bayesian is then recovered from the probabilistic DmBL. At last, it
is shown why DmBL is compliant with Lewis triviality.Comment: The revised version of "Definition of a Deterministic Bayesian
Logic". The formalism, proofs, and models have been enhanced and simplifie
Extension of Boolean algebra by a Bayesian operator; application to the definition of a Deterministic Bayesian Logic
This work contributes to the domains of Boolean algebra and of Bayesian
probability, by proposing an algebraic extension of Boolean algebras, which
implements an operator for the Bayesian conditional inference and is closed
under this operator. It is known since the work of Lewis (Lewis' triviality)
that it is not possible to construct such conditional operator within the space
of events. Nevertheless, this work proposes an answer which complements Lewis'
triviality, by the construction of a conditional operator outside the space of
events, thus resulting in an algebraic extension. In particular, it is proved
that any probability defined on a Boolean algebra may be extended to its
algebraic extension in compliance with the multiplicative definition of the
conditional probability. In the last part of this paper, a new bivalent logic
is introduced on the basis of this algebraic extension, and basic properties
are derived
Fuzzy Logic, Informativeness and Bayesian Decision-Making Problems
This paper develops a category-theoretic approach to uncertainty,
informativeness and decision-making problems. It is based on appropriate first
order fuzzy logic in which not only logical connectives but also quantifiers
have fuzzy interpretation. It is shown that all fundamental concepts of
probability and statistics such as joint distribution, conditional
distribution, etc., have meaningful analogs in new context. This approach makes
it possible to utilize rich conceptual experience of statistics. Connection
with underlying fuzzy logic reveals the logical semantics for fuzzy decision
making. Decision-making problems within the framework of IT-categories and
generalizes Bayesian approach to decision-making with a prior information are
considered. It leads to fuzzy Bayesian approach in decision making and provides
methods for construction of optimal strategies.Comment: 41 pages, LaTex, no figure
A Channel-Based Perspective on Conjugate Priors
A desired closure property in Bayesian probability is that an updated
posterior distribution be in the same class of distributions --- say Gaussians
--- as the prior distribution. When the updating takes place via a statistical
model, one calls the class of prior distributions the `conjugate priors' of the
model. This paper gives (1) an abstract formulation of this notion of conjugate
prior, using channels, in a graphical language, (2) a simple abstract proof
that such conjugate priors yield Bayesian inversions, and (3) a logical
description of conjugate priors that highlights the required closure of the
priors under updating. The theory is illustrated with several standard
examples, also covering multiple updating
Sampling First Order Logical Particles
Approximate inference in dynamic systems is the problem of estimating the
state of the system given a sequence of actions and partial observations. High
precision estimation is fundamental in many applications like diagnosis,
natural language processing, tracking, planning, and robotics. In this paper we
present an algorithm that samples possible deterministic executions of a
probabilistic sequence. The algorithm takes advantage of a compact
representation (using first order logic) for actions and world states to
improve the precision of its estimation. Theoretical and empirical results show
that the algorithm's expected error is smaller than propositional sampling and
Sequential Monte Carlo (SMC) sampling techniques.Comment: Appears in Proceedings of the Twenty-Fourth Conference on Uncertainty
in Artificial Intelligence (UAI2008
A Theoretical Framework for Context-Sensitive Temporal Probability Model Construction with Application to Plan Projection
We define a context-sensitive temporal probability logic for representing
classes of discrete-time temporal Bayesian networks. Context constraints allow
inference to be focused on only the relevant portions of the probabilistic
knowledge. We provide a declarative semantics for our language. We present a
Bayesian network construction algorithm whose generated networks give sound and
complete answers to queries. We use related concepts in logic programming to
justify our approach. We have implemented a Bayesian network construction
algorithm for a subset of the theory and demonstrate it's application to the
problem of evaluating the effectiveness of treatments for acute cardiac
conditions.Comment: Appears in Proceedings of the Eleventh Conference on Uncertainty in
Artificial Intelligence (UAI1995
Implementing a Library for Probabilistic Programming using Non-strict Non-determinism
This paper presents PFLP, a library for probabilistic programming in the
functional logic programming language Curry. It demonstrates how the concepts
of a functional logic programming language support the implementation of a
library for probabilistic programming. In fact, the paradigms of functional
logic and probabilistic programming are closely connected. That is, language
characteristics from one area exist in the other and vice versa. For example,
the concepts of non-deterministic choice and call-time choice as known from
functional logic programming are related to and coincide with stochastic
memoization and probabilistic choice in probabilistic programming,
respectively. We will further see that an implementation based on the concepts
of functional logic programming can have benefits with respect to performance
compared to a standard list-based implementation and can even compete with
full-blown probabilistic programming languages, which we illustrate by several
benchmarks. Under consideration in Theory and Practice of Logic Programming
(TPLP).Comment: Under consideration in Theory and Practice of Logic Programming
(TPLP
Safe Control under Uncertainty
Controller synthesis for hybrid systems that satisfy temporal specifications
expressing various system properties is a challenging problem that has drawn
the attention of many researchers. However, making the assumption that such
temporal properties are deterministic is far from the reality. For example,
many of the properties the controller has to satisfy are learned through
machine learning techniques based on sensor input data. In this paper, we
propose a new logic, Probabilistic Signal Temporal Logic (PrSTL), as an
expressive language to define the stochastic properties, and enforce
probabilistic guarantees on them. We further show how to synthesize safe
controllers using this logic for cyber-physical systems under the assumption
that the stochastic properties are based on a set of Gaussian random variables.
One of the key distinguishing features of PrSTL is that the encoded logic is
adaptive and changes as the system encounters additional data and updates its
beliefs about the latent random variables that define the safety properties. We
demonstrate our approach by synthesizing safe controllers under the PrSTL
specifications for multiple case studies including control of quadrotors and
autonomous vehicles in dynamic environments.Comment: 10 pages, 6 figures, Submitted to HSCC 201
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